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PHYSICAL REVIEW E VOLUME 60, NUMBER 6 DECEMBER 1999
High-frequency dynamics of wave localization
C W J Beenakker and K J H van Betnmelinstituut-Lorentz, Umversiteit Leiden, PO Box9506, 2300RA Leiden The Netherlands P W Brouwer
Laboratory of Atomic and Solid State Physics Cornell Umveisity, Ithaca, New York 14853 (Received 3 September 1999)
We study the effect of localization on the propagation of a pulse through a multimode disordered waveguide The correlator {«(ω1)ίί*(ω2)} of the transmitted wave amplitude u at two frequencies differmg by δω has for large δω the stretched exponential tail Kexp(— \Ιτοδω/2) The time constant TD = Li/D is given by the
diffusion coefficient D, even if the length L of the waveguide is much greater than the localization length ξ Localization has the effect of multiplymg the correlator by a frequency-mdependent factor exp(-L/2£), which disappears upon breakmg time-reversal symmetry [S1063-651X(99)50412-1]
PACS number(s) 42 25 Dd, 42 25 Bs, 72 15 Rn, 91 30 -f
The frequency spectrum of waves propagatmg through a random medium contams dynamical Information of mterest in optics [1], acoustics [2], and seismology [3] A fundamen-tal issue is how the phenomenon of wave localization [4] affects the dynamics The basic quantity is the correlation of the wave amplitude at two frequencies differmg by δω Α tecent microwave expenment by Genack et al [5] measured this correlation for a pulse transmitted through a waveguide with randomly positioned scatteiers The waves were not lo-cahzed in that expenment, because the length L of the wave-guide was less than the localization length ξ, so the cor-relatoi could be computed frorn the perturbation theory for diffusive dynamics [6] The charactenstic time scale in that regime is the time rD = L2/D it takes to diffuse (with
diffu-sion coefficient D) from one end of the waveguide to the other According to diffusion theory, foi large δω the cor-lelatoi decays °c exp( — \]TD δω/2) with time constant TD
What happens to the high-frequency decay of the cor-lelator if the waveguide becomes longer than the localization length7 That is the question addiessed in this Rapid Commu-nication Our prediction is that, although the conelatoi is suppressed by a factoi exp(—L/2£), the time scale for the decay lemains the diffusion time TD, even if diffusion is
only possible on length scales <L The exponential suppies-sion factoi disappears if time-reversal symmetry is broken (by some magneto-optical effect) Our analytical results are based on the formal equivalence between a frequency shift and an imagmary absorption rate, and aie supported by a numencal solution of the wave equation
We consider the propagation of a pulse through a disor-dered waveguide of length L In the frequency domam the transmission coefficient ίηη,(ω) gives the ratio of the
trans-mitted amplitude m mode n to the mcident amplitude m mode m (The modes aie normahzed to cany the same flux ) We seek the correlator €(δω) = (ίηη,(ω+ δω)^ιη(ω)) (The
brackets { } denote an average ovei the disoidei ) We as-sume that the (positive) frequency mciement δω is suffi-ciently small compared to ω that the mean fiee path / and the number of modes N in the waveguide do not vary appiecia-bly, and may be evaluated at the mean fiequency ω [7] We
also assume that 19>οΙω (with c the wave velocity) The localization length is then given by [8] ξ=(βΝ+2-β)1, with ß=l (2) m the presence (absence) of time-reversal symmetry For N^~ l the localization length is much greater than the mean fiee path, so that the motion on length scales below ξ is diffusive (with diffusion coefficient D)
Our appioach is to map the dynamic problem without absorption onto a staue problem with absorption [9] The mappmg is based on the analyticity of the transmission am-plitude tnm(w+iy), at complex frequency ω + iy with y
>0, and on the symmetry relation ίηηι(ω+ιγ) = ί*ιη( — ω
+ i y ) The pioduct of tiansmission amphtudes tnm(u>
+ z)tnm( — ω + ζ) is therefore an analytic function of z m the
upper half of the complex plane If we take z ieal, equal to ιδώ, we obtam the product of transmission amphtudes ίηηί(ω+ |·<5ω)ί*/η(ω — ^δω) considered above [the
differ-ence with ίη,,,(ω + 8ω)ί*ηι(ω) bemg statistically irrelevant
for δω<& ω] If we take z imagmary, equal to ι/2τα, we
ob-tain the transmission probabihty T= t ηη(ω + ι/2τα)\2 at
fre-quency ω and absorption täte 1/τα We conclude that the
conelatoi C can be obtamed from the ensemble average of T by analytic contmuation to imagmary absorption rate,
for \Ιτα-~>-ιδω (0
Two lemaiks on this mappmg (i) The effect of absorp-tion (with late l/r*) on €(δω) can be included by the sub-stitution l/ra—> — ιδω+ \Ιτ* This is of importance for com-panson with expenments, but here we will for simplicity ignoie this effect (n) Higher moments of the product C = fn m(a>+ ^δω)ί*η(ω—^δω) are related to higher moments
of 7 by (CP) = (TP) for 1/τα-+-ιδω This is not sufficient
to deteimme the entire probabihty distnbution P(C), because moments of the form (C^C*9) cannot be obtamed by ana-lytic contmuation [10]
To check the vahdity of this approach and to demonstiate how effective it is we consider bnefly the case Ν=ί Α disoideied smgle-mode waveguide is equivalent to a geom-etiy of parallel layeis with random vanations m composition and thickness Such a landomly stratified medium is studied
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R6314 BEENAKKER, van BEMMEL, AND BROUWER PRE 60
in seismology äs a model for the subsurface of the Earth [3] The correlator of the reflection amphtudes Κ(δω)
= (r(ω + δω)r*(ω)) has been computed in that context by White et al [11] (m the hmit L—>°o) Their icsult was
Γ
Jo (2)
The distribution of the reflection probability R= \r\ through an absorbmg smgle-mode waveguide had been studied many yeais earher äs a problem in radio-engineermg [12], with the result
(3)
One readily venfies that Eqs (2) and (3) are identical under the Substitution of IIτα by —ιδώ
In a similar way one can obtam the correlator of the trans-mission amphtudes by analytic continuation to imagmary ab-sorption rate of the mean transmission probability through an absorbmg waveguide The absorbmg problem for N= l was solved by Fieihkher, Pustilnik, and Yurkevich [13] That so-lution will not be considered furthei heie, since our mterest is in the multi-mode regime, lelevant for the rrucrowave ex-penments [5] The transmission probability in an absorbmg waveguide with N> l is given by [14]
(4) for absorption lengths ξΆ= \]θτα in the lange Ι^ξ^ξ The
length L of the waveguide should be §>/, but the relative magnitude of L and ξ is arbitrary Substitution of l/ra by
— ι δω gives the correlator
€(δω)=
N L smhV— ιτοδω
exp - ^ l ( 5 ) where rD = L2/D is the diffusion time The ränge of vahdity
of Eq (5) is ΙΙξ<^τ0δω<ζΙΙΙ, or equivalently D/£2<=5w
<Sc// In the diffusive regime, foi L<§£, the correlator (5) reduces to the known result [6] from pertuibation theory
Foi max(D/L2,D/£2)<l5<u<§c// the decay of the abso-lute value of the correlatoi is a stretched exponential,
2l
^777 (6)
In the locahzed regime, when ξ becomes smaller than L, the onset of this tau is pushed to higher frequencies, but it re-tams its functional form The weight of the tau is reduced by a factor exp(-L/2W/) m the presence of time-reversal sym-metry There is no reduction factoi if time-reveisal symsym-metry is broken
To lest our analytical fmdmgs we have camed out nu-mencal simulations The disordered medium is modeled by a two-dimensional squaie lattice (lattice constant a, length L, width W) The (relative) dielectnc constant ε fluctuates from site to site between l ± δε The multiple scattermg of a sca-lar wave Ψ (for the case /?= 1) is descnbed by discretizing
£ '
α + 0 J ί + l -2 ο" 5 -3 -4 1 0°o, N=5 0 L/l=18 l • L/l=36 2 α L/l=72 4 1 ι l . 2 4 6 8 10 (TDc5co)'/2FIG l Frequency dependence of the loganthm of the absolute value of the correlator €(δω) The data pomts follow from a nu-mencal Simulation for N=5, the solid curve is the analytical high-frequency result (6) for N^>1 (with ß=l) The decay of the cor-relator is given by the diffusive time constant rD=L2/D even if the length L of the waveguide is greater than the localization length ξ
= 61 The offset of about 0 6 between the numencal and analytical results is probably a fimte-N effect
the Helmholtz equation [V2 + (<u/c)2e]\l'' = 0 and computmg the üansmission matrix usmg the lecursive Gieen function techmque [15] The mean free path / is determmed fiom the aveiage transmission probability (Triit) = A ' ( l + L / / ) ~1 in the diffusive regime [8] The conelator C is obtamed by averagmg ίηιη(ω+δω)ί*ιη(ω) over the mode indices n,m
and over different realizations of the disorder We choose w2 = 2(c/a)2, <Se = 04, leadmg to 1 = 22 la The width W = l l a is kept flxed (correspondmg to N = 5 ) , while the length L is vaned in the lange (400-1600)« These waveguides aie well in the locahzed legime, Ll ξ ι anging from 3 to 12 A laige numbei (some l O4-l O5) of realizations were needed to average out the statistical fluctuations, and this restricted oui simulations to a relatively small value of W Foi the same reason we had to hmit the lange of δω in the data set with the laigest L
Results for the absolute value of the correlatoi are plotted m Fig l (data pomts) and are compaied with the analytical high-fiequency prediction for Λ/>1 (solid cuive) We see from Fig l that the correlators for different values of Ll ξ conveige for large δω to a curve that lies somewhat above the theoretical prediction The offset is about 0 6, and could be easily explamed äs an O( l) uncei tainty in the exponent in Eq (1) due to the fact that N is not > l m the Simulation Regardless of this offset, the Simulation confiims both ana-lytical predictions The stretched exponential decay <χεχρ(-Λ/τ£><5ω/2) and the exponential suppression factor exp(-L/2£) We emphasize that the time constant TD
= L2/D of the high-frequency decay is the diffusion time foi f/ze entire length L of the waveguide, even though the local-ization length ξ is up to a factoi of 12 smallei than L
We can summanze oui Undings by the Statement that the correlatoi of the transmission amphtudes factorizes in the high-fiequency regime €^/ι(δω)/2(ξ) The fiequency
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PRE 60 HIGH-FREQUENCY DYNAMICS OF WA VE LOCALIZATION R6315
Locahzation has no effect on fl, but only on /2 We can
contrast this factonzation with the high-frequency asymptot-ics K—>/3( δω) of the correlator of the reflection amphtudes In the corresponding absorbing pioblem the high-fiequency regime conesponds to an absorption length smaller than the locahzation length, so it is obvious that K becomes mdepen-dent of ξ m that regime The factonzation of C is less
obvi-ous Smce the locahzed regime is accessible expenmentally [16], we believe that an expeumental lest of our piediction should be feasible
Discussions with M Buttiker, L I Glazman, K A Matveev, M Pustilmk, and P G Silvestrov are gratefully acknowledged This work was supported by the Dutch Sci-ence Foundation NWO/FOM
[1] B A van Tiggelen, m Diffuse Waves m Complex Media, edited by J-P Fouque, NATO Science Senes C531 (Kluwer, Dor-drecht, 1999)
[2] R L Weavei, Phys Rev B 49, 5881 (1994)
[3] B White, P Sheng, and B Nair, Geophysics 55, 1158 (1990) [4] Scattermg and Locahzation of Classical Waves m Random
Media, edited by P Sheng (World Scientific, Smgapore,
1990)
[5] A Z Genack, P Sebbah M Stoytchev, and B A van Tiggelen, Phys Rev Lett 82, 715 (1999)
[6] R Berkovits and S Feng, Phys Rep 238, 135 (1994) [7] The length / = adlir differs from the transport mean free path lu
by a dimensionahty-dependent numencal coefficient ad
= 2,77/2,4/3 foi d =1,2,3 The diffusion coefficient is D
= cla/d
[8] CWJ Beenakker, Rev Mod Phys 69, 731 (1997)
[9] VI Klyatskm and A I Saichev, Usp Fiz Nauk 162, 161 (1992) [Sov Phys Usp 35, 231 (1992)]
[10] This is a complication of the transmission problem The reflec-tion problem is simpler, because the (approximate) umtanty of
the reflection matnx r provides additional Information on the distnbution of the correlator of the reflection amphtudes The mapping between the dynamic and absorbing problems has been used recentiy to calculate the entire distnbution of the eigenvalues of r(tü + \δω)τ\ω— ^δω) in the hmit δω—>0 S A Ramaknshna and N Kumar, e-pnnt cond-mat/9906098, C W J Beenakker and P W Brouwer, e-prmt cond-mat/9908325
[11] B White, P Sheng, Z Q Zhang, and G Papamcolaou, Phys Rev Lett 59, 1918 (1987)
[12] V N Tutubalm, Radiotekh Elektron 16, 1352 (1971) [Radio Eng Electron Phys 16, 1274 (1971)], W Kohler and G C Papamcolaou, SIAM (Soc Ind Appl Math) J Appl Math 30, 263 (1976)
[13] V Freihkher, M Pustilmk, and I Yurkevich, Phys Rev Lett 73, 810 (1994)
[14] P W Brouwer, Phys Rev B 57, 10 526 (1998)
[15] H U Baranger, D P DiVmcenzo, R A Jalabert, and A D Stone, Phys Rev B 44, 10 637 (1991)
